If you blow across the top of a bottle, the air particles inside are set into vibrational motion. This is referred to as air column resonance. For a long cylindrical column, there are clear mathematical relationships associated with the standing wave pattern, the frequency, and the wavelength. This Concept Builder focuses on air columns in which one end of the column is open to the surrounding atmosphere and the other end is closed off. These are called closed-end air columns.
Name That Harmonic: Closed-End Air Columns - Questions 2 Help
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Identify the standing wave pattern for a closed-end air column that is vibrating with a third harmonic wave pattern.
When an air column resonates with one of its harmonic frequencies, a standing wave pattern is formed in the column. Such a pattern has nodes - points that don't move - and antinodes - points that vibrate wildly. Nodes are located at the closed end since air is unable to vibrate in and out of the tube at that end. Antinodes are located at the open ends. There are a number of standing wave patterns that could form. In this question, you must identify the pattern associated with the third harmonic. The standing wave pattern for the first harmonic or fundamental frequency displays the longest possible wavelength of all the harmonics. Given the rule that there are vibrational nodes at the closed end and antinodes at the open-end, the longest wavelength would have one-quarter of a wavelength. Each successive harmonic after the first harmonic would have an additional node and antinode.
Odd Harmonics Only
There is one-fourth of a wave present in the pattern for the first harmonic. The next highest harmonic includes an extra one-half of a wave in the pattern. That means there is 3/4-ths of a wave present in the pattern. This leads to a pattern with one-third the wavelength of the first harmonic and three times the frequency. We call this the third harmonic since it is three times the frequency of the first harmonic. The next highest harmonic after the third harmonic has another one-half wave in the pattern for a total of 5/4-ths of a wave. It has one-fifth the wavelength of the first harmonic and five times the frequency. We call this the fifth harmonic since it has five times the frequency of the first harmonic. Applying the same reasoning, we would conclude that even-numbered harmonics do not exist for closed-end air columns.
Picking the Correct Pattern
As mentioned above, the third harmonic has 3/4-ths of a wave in the pattern. It begins with a node at the closed end and an antinode at the open end. You will need to tap through the choices until you find a pattern that meets these criteria.
Try this link to The Physics Classroom Tutorial for more help with the concept of standing wave patterns in Closed-end air columns:
Closed-End Air Columns